Theorem tpid2 4716

Description: One of the three elements of an unordered triple. (Contributed by NM, 7-Apr-1994.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)

Hypothesis

Ref	Expression
tpid2.1	⊢ 𝐵 ∈ V

Assertion

Ref	Expression
tpid2	⊢ 𝐵 ∈ {𝐴, 𝐵, 𝐶}

Proof of Theorem tpid2

Step	Hyp	Ref	Expression
1		eqid 2737	. . 3 ⊢ 𝐵 = 𝐵
2	1	3mix2i 1333	. 2 ⊢ (𝐵 = 𝐴 ∨ 𝐵 = 𝐵 ∨ 𝐵 = 𝐶)
3		tpid2.1	. . 3 ⊢ 𝐵 ∈ V
4	3	eltp 4634	. 2 ⊢ (𝐵 ∈ {𝐴, 𝐵, 𝐶} ↔ (𝐵 = 𝐴 ∨ 𝐵 = 𝐵 ∨ 𝐵 = 𝐶))
5	2, 4	mpbir 230	1 ⊢ 𝐵 ∈ {𝐴, 𝐵, 𝐶}

Syntax hints:   ∨ w3o 1085   = wceq 1540   ∈ wcel 2105  Vcvv 3441  {ctp 4575

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2708

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-tru 1543  df-ex 1781  df-sb 2067  df-clab 2715  df-cleq 2729  df-clel 2815  df-v 3443  df-un 3902  df-sn 4572  df-pr 4574  df-tp 4576

This theorem is referenced by:  wrdl3s3  14749  wwlks2onv  28427  elwwlks2ons3im  28428  umgrwwlks2on  28431  s3rn  31328  cyc3evpm  31525  sgnsf  31537  sgncl  32611  signsw0glem  32638  signsw0g  32641  signswmnd  32642  signswrid  32643  prodfzo03  32689  circlevma  32728  circlemethhgt  32729  hgt750lemg  32740  hgt750lemb  32742  hgt750lema  32743  hgt750leme  32744  tgoldbachgtde  32746  tgoldbachgt  32749  kur14lem7  33279  brtpid2  33769  rabren3dioph  40840  fourierdlem102  43986  fourierdlem114  43998  etransclem48  44060